How to calculate APR

There is no formula, and that is the point

You cannot rearrange an equation and read the APR off one side. Appendix J to Regulation Z defines it implicitly: the APR is the rate that makes a particular equation balance, and finding it requires solving numerically.

This surprises people who expect something like “APR = rate + fees ÷ term”. Approximations of that shape are always wrong, sometimes by enough to matter.

The equation

The APR is the rate at which the payments you are obligated to make, discounted back to closing, equal the money you actually received:

amount financed = Σ  payment / (1 + i)^n

where i is the rate per unit period and n runs over every scheduled payment. For a monthly loan the unit period is one month, and the disclosed APR is i × 12.

Note what is on each side. The payments are computed on the full note amount at the note rate — the lender bills you on the whole loan. The amount financed is smaller, because the prepaid finance charges came out of the proceeds. Repaying the whole while receiving less than the whole is precisely why the APR exceeds the note rate.

Step by step

1. Total the prepaid finance charges. Everything the lender imposes as a condition of the credit: discount points, origination, underwriting, processing, document preparation, mortgage insurance, prepaid interest. Third- party services you could shop for — appraisal, title, settlement — normally stay out.

2. Subtract them from the loan amount. That is the amount financed. Lender credits work the other way and can push it above the loan amount.

3. Build the payment stream from the note: the ordinary amortizing payment at the note rate over the full term, plus any balloon.

4. Solve for the rate that discounts that stream back to the amount financed. Start with the note rate, which is always too low, and search upward.

5. Multiply by the number of unit periods in a year. Twelve, for a monthly loan.

How the solving is actually done

Since the present value falls monotonically as the rate rises, the search is straightforward and cannot fail. Bisection is the dependable choice: bracket the answer between a rate that is clearly too low and one that is clearly too high, then halve the interval repeatedly. A few dozen iterations exhaust the precision of the arithmetic.

Newton’s method converges faster and is what most production software uses, but it can wander on flat segments of the curve — which is exactly where loans with heavy lender credits live. This site uses bisection: for a calculation that runs in microseconds, reliability is worth more than speed.

The implementation behind our calculator uses bisection, and it is the same code that renders the numbers you see on these pages.

Accuracy and tolerance

Regulation Z does not demand infinite precision. For a regular transaction, a disclosed APR is accurate if it falls within one-eighth of one percentage point of the correct figure — a wider band for irregular transactions.

So a calculator disagreeing with your Loan Estimate in the third decimal place means nothing. A disagreement of a quarter point means one of you is wrong about which fees are finance charges — and that is usually the interesting question.

What this method does not cover

The calculation above assumes equal payments and a regular first period. Real disclosures adjust for:

• odd days between closing and the first payment;
• adjustable rates, where the APR is built from a composite rate using the fully-indexed rate for the periods after the initial one;
• irregular payment streams — interest-only periods, step payments, seasonal schedules — which need the general equation rather than the annuity shortcut.

For fixed-rate mortgages and ordinary installment loans, the simple version matches lender disclosures comfortably inside the tolerance.